dgehrd.f

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      SUBROUTINE dgehrd( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEHRD reduces a real general matrix A to upper Hessenberg form H by
*  an orthogonal similarity transformation:  Q' * A * Q = H .
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) INTEGER
*          It is assumed that A is already upper triangular in rows
*          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
*          set by a previous call to DGEBAL; otherwise they should be
*          set to 1 and N respectively. See Further Details.
*          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the N-by-N general matrix to be reduced.
*          On exit, the upper triangle and the first subdiagonal of A
*          are overwritten with the upper Hessenberg matrix H, and the
*          elements below the first subdiagonal, with the array TAU,
*          represent the orthogonal matrix Q as a product of elementary
*          reflectors. See Further Details.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
*          zero.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length of the array WORK.  LWORK >= max(1,N).
*          For optimum performance LWORK >= N*NB, where NB is the
*          optimal blocksize.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  The matrix Q is represented as a product of (ihi-ilo) elementary
*  reflectors
*
*     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a real scalar, and v is a real vector with
*  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
*  exit in A(i+2:ihi,i), and tau in TAU(i).
*
*  The contents of A are illustrated by the following example, with
*  n = 7, ilo = 2 and ihi = 6:
*
*  on entry,                        on exit,
*
*  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
*  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
*  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
*  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
*  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
*  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
*  (                         a )    (                          a )
*
*  where a denotes an element of the original matrix A, h denotes a
*  modified element of the upper Hessenberg matrix H, and vi denotes an
*  element of the vector defining H(i).
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            NBMAX, LDT
      parameter( nbmax = 64, ldt = nbmax+1 )
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, LDWORK, LWKOPT, NB, NBMIN,
     $                   NH, NX
      DOUBLE PRECISION   EI
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   T( LDT, NBMAX )
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgehd2, dgemm, dlahrd, dlarfb, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      info = 0
      nb = min( nbmax, ilaenv( 1, 'DGEHRD', ' ', n, ilo, ihi, -1 ) )
      lwkopt = n*nb
      work( 1 ) = lwkopt
      lquery = ( lwork.EQ.-1 )
      IF( n.LT.0 ) THEN
         info = -1
      ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
         info = -2
      ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -5
      ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
         info = -8
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGEHRD', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
*
      DO 10 i = 1, ilo - 1
         tau( i ) = zero
   10 CONTINUE
      DO 20 i = max( 1, ihi ), n - 1
         tau( i ) = zero
   20 CONTINUE
*
*     Quick return if possible
*
      nh = ihi - ilo + 1
      IF( nh.LE.1 ) THEN
         work( 1 ) = 1
         RETURN
      END IF
*
*     Determine the block size.
*
      nb = min( nbmax, ilaenv( 1, 'DGEHRD', ' ', n, ilo, ihi, -1 ) )
      nbmin = 2
      iws = 1
      IF( nb.GT.1 .AND. nb.LT.nh ) THEN
*
*        Determine when to cross over from blocked to unblocked code
*        (last block is always handled by unblocked code).
*
         nx = max( nb, ilaenv( 3, 'DGEHRD', ' ', n, ilo, ihi, -1 ) )
         IF( nx.LT.nh ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            iws = n*nb
            IF( lwork.LT.iws ) THEN
*
*              Not enough workspace to use optimal NB:  determine the
*              minimum value of NB, and reduce NB or force use of
*              unblocked code.
*
               nbmin = max( 2, ilaenv( 2, 'DGEHRD', ' ', n, ilo, ihi,
     $                 -1 ) )
               IF( lwork.GE.n*nbmin ) THEN
                  nb = lwork / n
               ELSE
                  nb = 1
               END IF
            END IF
         END IF
      END IF
      ldwork = n
*
      IF( nb.LT.nbmin .OR. nb.GE.nh ) THEN
*
*        Use unblocked code below
*
         i = ilo
*
      ELSE
*
*        Use blocked code
*
         DO 30 i = ilo, ihi - 1 - nx, nb
            ib = min( nb, ihi-i )
*
*           Reduce columns i:i+ib-1 to Hessenberg form, returning the
*           matrices V and T of the block reflector H = I - V*T*V'
*           which performs the reduction, and also the matrix Y = A*V*T
*
            CALL dlahrd( ihi, i, ib, a( 1, i ), lda, tau( i ), t, ldt,
     $                   work, ldwork )
*
*           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
*           right, computing  A := A - Y * V'. V(i+ib,ib-1) must be set
*           to 1.
*
            ei = a( i+ib, i+ib-1 )
            a( i+ib, i+ib-1 ) = one
            CALL dgemm( 'No transpose', 'Transpose', ihi, ihi-i-ib+1,
     $                  ib, -one, work, ldwork, a( i+ib, i ), lda, one,
     $                  a( 1, i+ib ), lda )
            a( i+ib, i+ib-1 ) = ei
*
*           Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
*           left
*
            CALL dlarfb( 'Left', 'Transpose', 'Forward', 'Columnwise',
     $                   ihi-i, n-i-ib+1, ib, a( i+1, i ), lda, t, ldt,
     $                   a( i+1, i+ib ), lda, work, ldwork )
   30    CONTINUE
      END IF
*
*     Use unblocked code to reduce the rest of the matrix
*
      CALL dgehd2( n, i, ihi, a, lda, tau, work, iinfo )
      work( 1 ) = iws
*
      RETURN
*
*     End of DGEHRD
*
      END